Solutions and Improved Perturbation Analysis for the Matrix Equation X-A*X-pA=Q (p>0)‎

Abstract EN

The nonlinear matrix equation X-A*X-pA=Q with p>0 is investigated.

We consider two cases of this equation: the case p≥1 and the case 0

In the case p≥1, a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained.

A perturbation estimate for the positive definite solution is derived.

Explicit expressions of the condition number for the positive definite solution are given.

In the case 0

A new backward error of an approximate solution to the unique positive definite solution is obtained.

The theoretical results are illustrated by numerical examples.